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In Western popular music, drums and percussion are an important means to emphasize and shape the rhythm, often defining the musical style. If computers were able to analyze the drum part in recorded music, it would enable a variety of rhythm-related music processing tasks. Especially the detection and classification of drum sound events by computational methods is considered to be an important and challenging research problem in the broader field of Music Information Retrieval. Over the last two decades, several authors have attempted to tackle this problem under the umbrella term Automatic Drum Transcription (ADT). This paper presents a comprehensive review of ADT research, including a thorough discussion of the task-specific challenges, categorization of existing techniques, and evaluation of several state-of-the-art systems. To provide more insights on the practice of ADT systems, we focus on two families of ADT techniques, namely methods based on Non-negative Matrix Factorization and Recurrent Neural Networks. We explain the methods' technical details and drum-specific variations and evaluate these approaches on publicly available datasets with a consistent experimental setup. Finally, the open issues and under-explored areas in ADT research are identified and discussed, providing future directions in this field.